Chinese immigrants to Southeast Asia, the US, Europe, and even Taiwan in the 19th and early 20th centuries seem to have favoured identifying as people of Tang (唐人) while Chinese people today identify as people of Han (漢人). This is evident especially in Hokkien / Hakka / Cantonese, where the Chinese language is the Tang language (唐語), Chinatown is the Street of Tang People (唐人街), and China itself is the Mountains of Tang (唐山 as in 唐山過南洋).
I can understand that they didn't want to identify as Qing given the Manchu angle, but the selection of Tang seems odd. Han was already a semi-official designation, and the last non-Manchu dynasty was the Ming, so why didn't they go with Ming? Secret societies across the Overseas Chinese community like the Tiandihui were at least formally devoted to Overthrow Qing and Restore Ming (反清復明), so it seemed there was already broad popularity for the Ming. Tang just seems both too far away for anyone to relate to, and yet not old enough that a claim for Tang to be the "beginning" of Chineseness could be made (especially considering how religiously and culturally diverse Tang was).
One thing to consider is that Chinese emigrants during the 19th and early 20th century were largely from the Southeast, particularly Guangdong and Fujian. These areas were not as integrated into more northern Chinese rule during the Han dynasty and were more Sinified during the Tang Dynasty - in fact many Cantonese speakers still call themselves Tang people. This may be changing with the growing numbers of mainland Mandarin speakers in overseas Chinese communities.